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I want to estimate delta, vega and gamma for a basket option. This option is a European Call option. The underlying is $S=\omega_1 S_1 +\omega_2 S_2$

Where:

$S1$ = stock price of asset 1

$S2$ = stock price of asset 2

$\omega_1 $ and $\omega_2$ are the weights

It is easy to compute $\nu_{S_1} $ and $\nu_{S_2}$ but my question is about the definition of $\nu_{S}$

I started computing $\nu_{S}$ as $\omega_1 \nu_{S_1} +\omega_2 \nu_{S_2}$ but then I realized that maybe this is not the correct way to compute it.

My second option was to estimate a number $\theta$ with the following property: "A variation on $\theta$ of each of the stock's volatilities produces a variation on the SD of S of magnitude $\epsilon$", then, compute vega as $\nu= (V(\sigma_1+\theta, \sigma_2 +\theta)-V(\sigma_1, \sigma_2))/ \epsilon$ for a small $\epsilon$ value

Where V represent the option price and $\sigma_i$ the volatlity of the ith asset.

I want to know if one of these methods is correct.

I am new at pricing options so every information or recomendation will be useful for me.

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